What is the Sum of Discrete Sinusoids?

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Homework Help Overview

The discussion revolves around the sum of two discrete sinusoidal functions, specifically sin[8*\pi*n/5] and sin[12*\pi*n/5], with the goal of demonstrating that their sum is zero for integer values of n.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants present numerical evaluations of the sinusoidal functions for various integer values of n, suggesting a pattern that may lead to the conclusion of the sum being zero. Some participants question the geometric interpretation of the functions on the unit circle, particularly regarding the relationship between their y-components.

Discussion Status

The discussion includes attempts to analyze the functions both numerically and geometrically. Some participants have offered hints regarding the relationship between the two sinusoidal functions, while others explore the implications of their positions on the unit circle. There is no explicit consensus, but several productive lines of reasoning are being explored.

Contextual Notes

Participants are working under the assumption that the sum of the two sinusoidal functions can be shown to be zero, and there is a focus on finding an analytic solution. The discussion reflects a learning environment where assumptions and interpretations are being questioned.

Superman1271
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Homework Statement



Hi Everyone, I am trying to show why the given sum is zero. I am pretty sure it is zero.




Homework Equations



sin[8*\pi*n/5]+sin[12*\pi*n/5]

n is an integer.

The Attempt at a Solution



n----sin[8*\pi*n/5]----sin[12*\pi*n/5]

0 ---- 0------------------------------ 0

1 ---- -0.9511------------------------ 0.9511

2 ---- -0.5878------------------------ 0.5878

3 ---- 0.5878------------------------ -0.5878

4 ---- 0.9511------------------------ -0.9511

5 ---- 0------------------------------- 0

I am looking for an analytic solution thank you.
 
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Superman1271 said:

Homework Statement



Hi Everyone, I am trying to show why the given sum is zero. I am pretty sure it is zero.

Homework Equations



sin[8*\pi*n/5]+sin[12*\pi*n/5]

n is an integer.

The Attempt at a Solution



n----sin[8*\pi*n/5]----sin[12*\pi*n/5]

0 ---- 0------------------------------ 0

1 ---- -0.9511------------------------ 0.9511

2 ---- -0.5878------------------------ 0.5878

3 ---- 0.5878------------------------ -0.5878

4 ---- 0.9511------------------------ -0.9511

5 ---- 0------------------------------- 0

I am looking for an analytic solution thank you.

It's because the point between those two numbers is 10*pi*n/5=2*pi*n. Enough of a hint?
 
Last edited:
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The point exactly half way between those two numbers is 2pi n.
 
So on the unit circle the halfway point is always along the positive x-axis. And if that is halfway, the y components will always be the negative of each other?

Also how did you realize it was exactly half way?
 
Superman1271 said:
So on the unit circle the halfway point is always along the positive x-axis. And if that is halfway, the y components will always be the negative of each other?

Also how did you realize it was exactly half way?

Thinking about where the points for various n lie on the unit circle is a good way.
 
Ok, Thanks =]
 

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